Axioms Definition and 181 Threads

Hi there, In most books that I saw, the set of real numbers under the usual sum and product is considered as a Field and say that's by the field axioms. But I have surprised when I have seen, it is a theorem. The question, are these axioms? or can they be proved? Thank you very much.

Homework Statement I've been steadily working through Apostol's Calculus, Volume 1 and trying to prove all of the theorems and solve all of the exercises, but I skipped a few theorems and exercises in the Field Axiom section (I 3.2 & I 3.3). The lack of continuity irks me, so I sometimes flip...

Homework Statement For each of the following definitions for a * b and a given set, determine which of the axioms G0, G1, G2, G3 are satisfied by a * b. In which cases do we obtain a group? 1) a - b on the set Z 2)a + b - ab on the set R | {1} 3)ab on {2^n | n in Z} Homework...

Thanks in advance. 1st day at calculus teacher wants proofs. They seem rudimentary but I've never done them and he doesn't help so I'm hoping someone here could please. These are the axioms: Addition: For a, b, and c taken from the real numbers A1: a+b is a real number also (closure) A2...

Homework Statement Prove that \vec0 + \vec{A} = \vec{A} The Attempt at a Solution \vec{A} + (-\vec{A}) = \vec0 \vec0 + \vec{A} = \vec{A} + (-\vec{A}) + \vec{A} = \vec{A} + \vec{A} + (- \vec{A}) \vec0+ \vec{A} = \vec{A} + \vec0 = \vec{A} I find this last line somewhat unconvincing...

Homework Statement Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms. Homework Equations N/A The Attempt at a Solution I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove...

Homework Statement Show that the axiom \vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C} holds for polynomials of the form a_0 + a_1 x + a_2 x^2 The Attempt at a Solution I'm pretty new to writing proofs for linear algebra so my first question is should I be treating the...

What are the equations from which all of GR can be derived? Obviously one of the equations is Einstein's Field Equation: G^{\alpha\beta}=8\pi T^{\alpha\beta}. I would also guess that you would need the Euler-Lagrange Equations: -\frac{d}{d\sigma}(\frac{\partial L}{\partial...

Homework Statement Prove that P(A \cap B)≥1-P(\bar{A})-P(\bar{B}) for all A, B \subseteq Susing only these axioms: 1) 0 \leq P(A) \leq 1, for any event A \subseteq S 2) P(S) = 1 3) P(A \cup B) = P(A) + P(B) if and only if P(A \cap B) = 0Homework Equations None. The Attempt at a Solution My...

Does anyone know of any good books on Axioms. Such as how was ZF(C) came up with and why it is that the general consensus is that it is complete.

I came across this question: http://imageshack.us/m/3/4510/axiomsq.png which I'm confused about. I know what the axioms of probability are but how would I use them to "derive" that result? I could illustrate why P(A∪B) = P(A) + P(B) - P(A∩B) on a Venn diagram but I have no idea how to use the...

Homework Statement I have quite a few problems that I believe I answered correctly, but here is one of them: 1. Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows: (x1, y1, z1) + (x2, y2, z2) = (x1+ x2...

Anybody tired of those long, tedious proofs in ZFC? Anybody tired of those annoying counterexamples to beautiful results? Want to see a new kind of mathematics where everything works fine? Try Falso: http://estatis.coders.fm/falso/

Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics? Thanks...

I propose this set of axioms and ask you to be bring arguments for it and arguments against it. What other axioms would you choose instead of the ones I wrote below ? 1. STATE DESCRIPTION: All physical states of a quantum system are described mathematically by a set at most countable of...

Can someone please help me?

Hi, I am trying to work through Finite Dimensional Vector Spaces by Halmos, and I am having some difficulty with the first problem on page two (the specific problem is included below). The last class I took involving formal proofs was linear algebra about 8 years ago, and I am very rusty, but I...

What are the axioms of quantum Physics? Does time/space considered fundamental quantities that considered to exist in nature when formulating quantum theory?

I've looked around but haven't found anyway to prove 2+2=4. I'm pretty sure you need to use field axioms, but I just haven't found it yet. Is there a way to do it? Like showing a+a=2a? Or a+b=c? Like 1+1=2. Something like that. Thanks!

Homework Statement Consider the collection F of all real numbers of the form x+y√2, where x and y are rational numbers. Prove (by direct verification) that F satisfies all the field axioms (just like R) under the usual addition and multiplication. Homework Equations Field axioms...

Homework Statement prove : 0 = -0 Homework Equations The Attempt at a Solution

1. The problem statement: Prove that 0\leq2x^2 - 3xy + 2y^2 2. These are the axioms we are permitted to use: 01) Exactly one of these hold: a<b, a=b, or b<a 02) If a<b, and b<c, then a<c 03) If a<b, then a+c < b+c for every c 04) If a<b and 0<c, then ac<bc. The Attempt at...

Analysis Help; proofs via axioms :) 1. The problem statement: Prove that for any real numbers a, b, c, (a+b+c)^2\leq3*(a^2 +b^2+c^2) 2. These are the axioms we are permitted to use: 01) Exactly one of these hold: a<b, a=b, or b<a 02) If a<b, and b<c, then a<c 03) If a<b, then...

Homework Statement Using only the ring axioms, prove that in a general ring (R, +,X) aX (x-z) = (aXx)- (aXz) where all a,x,z are elements of R Homework Equations Group axiom 3: G3= There is an inverse for each element g^-1 *g =e Ring axiom 3: R3= Two distributive laws...

Homework Statement Show that a+b = b+a follows from the other ring axioms. Homework Equations a + 0 = 0 + a = a (?) The Attempt at a Solution I know this is probably a simple algebraic manipulation, probably with the distributive law, but I don't know how to start! I'm sure I need...

Homework Statement If we define V as the set of vectors in R^2 with vector addition defined as it normally is, but scalar multiplication defined to be k(a,b)=(k2a,k2b), then V is not a vector space. Find all axioms that fail (and explain why they fail). Homework Equations Axioms: 1. If u and...

Since I can't copy and paste from maple into this message w/out losing formatting, I attached a pdf with all the work. I am having trouble proving axiom 1 of two general magic square matrices added together; plus, I am not sure if my set notation is entirely correct.

I had a question about axioms. Assuming I understand this correctly, axioms can neither be proven nor disproven; they are self-evident definitions that we have made to simplify math. So someone (with a strong religious motivation I might add) said that axioms are based on faith. You can't...

Hello everyone. I was wondering wether people think that mathematical conclusions can be provisional? I know about conjectures, but are they really part of maths? Finally, my definition of an axiom is: self evident and accepted Do you people think that there are better definitions than...

Hi. please anyone help me with vector spaces and the way to prove the axioms. like proving that (-1)u=-u in a vector space.

Homework Statement We're asked to prove that a few constructions of the sets a,b are themselves sets, stating which axioms we use to do so. a) a\b b)the function f:a->b c)the image of f Homework Equations The following standard definitions of axioms of construction...

what is the convention you adhere to when it comes to natural numbers? for example there is a long standing debate about 0... should we define \mathbb N = \{0,1,2,...\} or instead \mathbb N = \{1,2,3,...\} and more about this, considering Peano's Axioms than we could choose \mathbb N...

Homework Statement "Prove that Axiom 10[, the existence of identity in a linear space,] can be deduced from the other axioms." Homework Equations Now, I know that these axiom numberings are fairly arbitrary, but I'll put them in anyways for easy reference. I'm listing them word for word...

I was reading "Principles of Quantum Mechanics" - Shankar, and I'm having trouble understanding the inner product. Can someone help me or link me to a site that explains it? The axioms of the inner product are 1. \langle V|W\rangle = \langle W|V\rangle^* 2. \langle V|V\rangle \geq 0\ \...

Are the axioms of math subjective? If they are, then logically all the formal consequences that flow from them are also subjective. Thus there can be no objective mathematical facts. Someone said this: Someone then replied this: But then later made the contrasting statement: A...

Homework Statement Prove that if 0 < a < b, then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Axioms (Properties), courtesy of Wikipedia: Addition: P1: For all a, b, and c in F, a + (b + c) = (a + b) + c P2: There exists an element of F, called the additive identity...

If you have a > b and c \geq d, do you have a + c > b + d?

Hi Can anyone help me define the axioms and theorems and what the differences are? I know axioms are suppose to be statements that are considered true based on logic (ex. x+y=y+x) but cannot be proven. Can someone explain why it can't be proven? Thanks

Homework Statement On R2, define the binary operators (x,y)+(u,v)=(x+u,y+v) (x,y)+(u,v)=(xu+yv,xv+yu) The set R2, along with these definitions of addition and multiplication, for the perplex numbers. (a) Show that \cdot: R2 \rightarrow R is associative...

1) For any x > 0 and 0 ≤ h < 1 we have (x + h)^2 ≤ x^2 + h(2x + 1). 2) For any x > 0 and p > 0 with x^2 < p there exists y > x with y^2 < p. Prove the following statements (only using the axioms for the real numbers). At each step say which axiom you use. The problems is that my professor...

hall, i need to prove by using the field axioms that: (-a)(-b)=ab, i think i know how to this but I'm very insecure with using those axioms cause i want to make sure I'm not using my intuition. i tried something like: (-a)(-b)=(-1)(a)(-1)b=(-1)(-1)(a)(b)=ab and i guess it's wrong...

Homework Statement determine if the space is a subspace testing both closure axioms. in R^2 the set of vectors (a,b) where ab=0 Homework Equations The Attempt at a Solution i just used the sum and product which are the closure axioms. But at the end how do you tell if the...

If a problem I'm doing asks to find V2 where V is a vector is it simply the dot product of the vector, or the cross product? The question: Which of the following sets of vectors v = {v1,...,vn} in Rn are subspaces of Rn (n>=3) iii) All v such that V2=V12 He proved it by saying...

Why are they called "axioms"? Shouldn't they be called "definitions"?

since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up. thanx

Homework Statement Let F be any field, and fix a є F. Equip the set V = F2 with two operations as follows. Define addition by (x, y)‡(x', y') := (x + x', y + y' − a), for all x, x', y, y' є F, and define the scalar multiplication by scalars by c * (x, y) := (cx, cy − ac + a), for all x...

Is the statement -1<1/0<1 decidable using the ordered field/real number axioms and first order logic? I have tried to prove that the statement is either true or false but have had no success since the axioms and theorems only make statements about objects that exist and do not give any clear way...

I am trying to shorten and generalize the the definition of a vector space to redefine it in such a way that only four axioms are required. The axioms must hold for all vectors u, v and w are in V and all scalars c and d. I believe the four would be: 1. u + v is in V, 2. u + 0 = u 3. u...

Self explanatory. The Cantor's theorem which am referring to is that the cardinality of the power set of any set is greater than that of the set.

My assignment is like this: 1.give an example of a space X and a subspace A of X s.t X satisifes Sep and A doesnt. 2.give an example of a continuous and onto function f:X->Y s.t X satisifies S1 but Y doesnt. 3.give an example of a continuous and onto function f:X->Y s.t X satisfies S2 and Y...